Post on 18-Dec-2015
Lorentz Centre 2 October, 2006
The Energy Spectrumof the
Atmosphere
Peter LynchUniversity College Dublin
Geometric & Multi-scale Methods for Geophysical Fluid Dynamics
Lorentz Centre, University of Leiden
Lorentz Centre 2 October, 2006
The Problem
• A complete understanding of the atmospheric energy spectrum remains elusive.
• Attempts using 2D and 3D and Quasi-Geostrophic turbulence theory to explain the spectrum have not been wholly satisfactory.
Lorentz Centre 2 October, 2006
Quasi-Geostrophic Turbulence
• The characteristic aspect ratio of the atmosphere is 100:1
L/H ~ 100
Lorentz Centre 2 October, 2006
Quasi-Geostrophic Turbulence
• The characteristic aspect ratio of the atmosphere is 100:1
L/H ~ 100
• Is quasi-geostrophic turbulence more like 2D or 3D turbulence?
Lorentz Centre 2 October, 2006
2D Vorticity Equation
• In 2D flows, the vorticity is a scalar:
• For non-divergent, non-rotating flow:
vx
uy
t
ux
vy
ddt
0
Lorentz Centre 2 October, 2006
2D Vorticity Equation
• If we introduce a stream function , we can write the vorticity equation as
• The velocity is
t
2 J ,2 0
(u,v) y
,x
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Quasi-Geostrophic Potential Vorticity
• In the QG formulation we seek to augment the 2D picture in two ways:
Lorentz Centre 2 October, 2006
Quasi-Geostrophic Potential Vorticity
• In the QG formulation we seek to augment the 2D picture in two ways:
– We include the effect of the Earth’s rotation.
Lorentz Centre 2 October, 2006
Quasi-Geostrophic Potential Vorticity
• In the QG formulation we seek to augment the 2D picture in two ways:
– We include the effect of the Earth’s rotation.
– We allow for horizontal divergence.
Lorentz Centre 2 October, 2006
Quasi-Geostrophic Potential Vorticity
• The equation of Conservation of Potential Vorticity is:
relative vorticity– f - planetary vorticity– h - fluid height
d
dt
fh
0
Lorentz Centre 2 October, 2006
Quasi-Geostrophic Potential Vorticity
• To derive a single equation for a single variable, we assume geostrophic balance:
• This allows us to relate the mass and wind fields.
V g
fkh k
Lorentz Centre 2 October, 2006
QGPV Equation• The Barotropic Quasi-Geostrophic Potential Vorticity
Equation is:
where .
t
2 F J ,2 x
0
F f0
2
gH
Lorentz Centre 2 October, 2006
QG Turbulence: 2D or 3D?
• 2D Turbulence– Energy & Enstrophy conserved– No vortex stretching
Lorentz Centre 2 October, 2006
QG Turbulence: 2D or 3D?
• 2D Turbulence– Energy & Enstrophy conserved– No vortex stretching
• 3D Turbulence– Enstrophy not conserved– Vortex stretching present
Lorentz Centre 2 October, 2006
QG Turbulence: 2D or 3D?
• 2D Turbulence– Energy & Enstrophy conserved– No vortex stretching
• 3D Turbulence– Enstrophy not conserved– Vortex stretching present
• QG Turbulence– Energy & Enstrophy conserved (like 2D)– Vortex stretching present (like 3D)
Lorentz Centre 2 October, 2006
QG Turbulence: 2D or 3D?
• The prevailing view has been that QG turbulence is more like 2D turbulence.
Lorentz Centre 2 October, 2006
QG Turbulence: 2D or 3D?
• The prevailing view has been that QG turbulence is more like 2D turbulence.
• The mathematical similarity of 2D and QG flows prompted Charney (1971) to conclude that an energy cascade to small-scales is impossible in QG turbulence.
Lorentz Centre 2 October, 2006
Inverse cascade to largest scales
Inverse cascade to isolated vortices
Lorentz Centre 2 October, 2006
Some Early Results
• Fjørtoft (1953) – In 2D flows, if energy is injected at an intermediate scale, more energy flows to larger scales.
Lorentz Centre 2 October, 2006
Some Early Results
• Fjørtoft (1953) – In 2D flows, if energy is injected at an intermediate scale, more energy flows to larger scales.
• Charney (1971) used Fjørtoft’s proofs to derive the conservation laws for QG turbulence.
Lorentz Centre 2 October, 2006
Some Early Results
• Fjørtoft (1953) – In 2D flows, if energy is injected at an intermediate scale, more energy flows to larger scales.
• Charney (1971) used Fjørtoft’s proofs to derive the conservation laws for QG turbulence.
• The proof used is really just a convergence requirement for a spectral representation of enstrophy (Tung & Orlando, 2003).
Lorentz Centre 2 October, 2006
2D Turbulence
• Standard 2D turbulence theory predicts:
– Upscale energy cascade from the point of energy injection (spectral slope –5/3)
Lorentz Centre 2 October, 2006
2D Turbulence
• Standard 2D turbulence theory predicts:
– Upscale energy cascade from the point of energy injection (spectral slope –5/3)
– Downscale enstrophy cascade to smaller scales (spectral slope –3)
Lorentz Centre 2 October, 2006
2D Turbulence
• Inverse Energy Cascade
• Forward Enstrophy Cascade
35
32
)(
kkE
E(k) 2
3k 3
Lorentz Centre 2 October, 2006
2D Turbulence
• Inverse Energy Cascade
• Forward Enstrophy Cascade
35
32
)(
kkE
E(k) 2
3k 3
What observational evidence do we have?
Lorentz Centre 2 October, 2006
Two Mexican physicists,José Luis Aragón andGerardo Naumis, haveexamined the patterns in van Gogh’s
Starry Night
Lorentz Centre 2 October, 2006
Two Mexican physicists,José Luis Aragón andGerardo Naumis, haveexamined the patterns in van Gogh’s
Starry Night
They found that the PDF of luminosity follows a Kolmogorov -5/3 scaling law.
See Plus e-zine for more information.
Lorentz Centre 2 October, 2006
Observational Evidence
• The primary source of observational evidence of the atmospheric spectrum remains (over 20 years later!) the study undertaken by Nastrom and Gage (1985)
[but see also the MOZAIC dataset].
• They examined data collated by nearly 7,000 commercial flights between 1975 and 1979.
• 80% of the data was taken between 30º and 55ºN.
Lorentz Centre 2 October, 2006
Observational Evidence
• No evidence of a broad mesoscale “energy gap”.
Lorentz Centre 2 October, 2006
Observational Evidence
• No evidence of a broad mesoscale “energy gap”.
• Velocity and Temperature spectra have nearly the same shape.
Lorentz Centre 2 October, 2006
Observational Evidence
• No evidence of a broad mesoscale “energy gap”.
• Velocity and Temperature spectra have nearly the same shape.
• Little seasonal or latitudinal variation.
Lorentz Centre 2 October, 2006
Observed Power-Law Behaviour
• Two power laws were evident:
• The spectrum has slope close to –(5/3) for the range of scales up to 600 km.
Lorentz Centre 2 October, 2006
Observed Power-Law Behaviour
• Two power laws were evident:
• The spectrum has slope close to –(5/3) for the range of scales up to 600 km.
• At larger scales, the spectrum steepens considerably to a slope close to –3.
Lorentz Centre 2 October, 2006
The Spectral “Kink”
• The observational evidence outlined above showed a kink at around 600 km– Surely too large for isotropic 3D effects?
Lorentz Centre 2 October, 2006
The Spectral “Kink”
• The observational evidence outlined above showed a kink at around 600 km– Surely too large for isotropic 3D effects?
• Nastrom & Gage (1986) suggested the shortwave –5/3 slope could be explained by another inverse energy cascade, from convective storm scales (after Larsen, 1982)
Lorentz Centre 2 October, 2006
The Spectral “Kink” (cont.)
• Lindborg & Cho (2001), however, could find no support for an inverse energy cascade at the mesoscales.
Lorentz Centre 2 October, 2006
The Spectral “Kink” (cont.)
• Lindborg & Cho (2001), however, could find no support for an inverse energy cascade at the mesoscales.
• Tung and Orlando (2002) suggested that the shortwave k^(-5/3) behaviour was due to a small downscale energy cascade from the synoptic scales.
Lorentz Centre 2 October, 2006
The Spectral Kink
• Tung and Orlando reproduced the N&G spectrum using QG dynamics alone. (They employed sub-grid diffusion.)
20
Lorentz Centre 2 October, 2006
The Spectral Kink
• Tung and Orlando reproduced the N&G spectrum using QG dynamics alone. (They employed sub-grid diffusion.)
• The NMM model also reproduces the spectral kink at the mesoscales when physics is included (Janjic, EGU 2006)
20
Lorentz Centre 2 October, 2006
No Physics With Physics
Where is the small scale energy in the observed spectrum coming from?
Atlantic case, NMM-B, 15 km, 32 Levels
(Thanks to Zavisa Janjic for this slide)
Lorentz Centre 2 October, 2006
An Additive Spectrum?• Charney (1973) noted the possibility of an additive
spectrum:
• Tung & Gkioulekas (2005) proposed a similar form:
E(k) Ak 3 Bk 5
3 Ck 2
E(k) Ak 3 Bk 5
3
Lorentz Centre 2 October, 2006
Current View of Spectrum
• Energy is injected at scales associated with baroclinic instability.
Lorentz Centre 2 October, 2006
Current View of Spectrum
• Energy is injected at scales associated with baroclinic instability.
• Most injected energy inversely cascades to larger scales (-5/3 spectral slope)
Lorentz Centre 2 October, 2006
Current View of Spectrum
• Energy is injected at scales associated with baroclinic instability.
• Most injected energy inversely cascades to larger scales (-5/3 spectral slope)
• Large-scale energy is lost through radiative dissipation & Ekman damping.
Lorentz Centre 2 October, 2006
Current Picture (cont.)
• It is likely that a small portion of the injected energy cascades to smaller scales.
Lorentz Centre 2 October, 2006
Current Picture (cont.)
• It is likely that a small portion of the injected energy cascades to smaller scales.
• At synoptic scales, the downscale energy cascade is spectrally dominated by the k^(-3) enstrophy cascade.
Lorentz Centre 2 October, 2006
Current Picture (cont.)
• Below about 600 km, the downscale energy cascade begins to dominate the energy spectrum.
Lorentz Centre 2 October, 2006
Current Picture (cont.)
• Below about 600 km, the downscale energy cascade begins to dominate the energy spectrum.
• The slope is evident at scales smaller than this.
k 5
3
Lorentz Centre 2 October, 2006
Current Picture (cont.)
• Below about 600 km, the downscale energy cascade begins to dominate the energy spectrum.
• The slope is evident at scales smaller than this.
• The slope is probably augmented by an inverse energy cascade from convective scales.
k 5
3
k 5
3
Lorentz Centre 2 October, 2006
Inverse Enstrophy Cascade?
• It is possible that a small portion of the enstrophy inversely cascades from synoptic to planetary scales.
Lorentz Centre 2 October, 2006
Inverse Enstrophy Cascade?
• It is possible that a small portion of the enstrophy inversely cascades from synoptic to planetary scales.
• We are unlikely, however, to find evidence of large-scale behaviour:– The Earth’s circumference dictates the
size of the largest scale.
k 3
Lorentz Centre 2 October, 2006
ECMWF Model Output
• The “kink” at mesoscales is not evident in the ECMWF model output.
k 5
3
Lorentz Centre 2 October, 2006
ECMWF Model Output
• The “kink” at mesoscales is not evident in the ECMWF model output.
• Excessive damping of energy is likely to be the cause.
(Thanks to Tim Palmer & Glenn Shutts for the following figures)
k 5
3
Lorentz Centre 2 October, 2006
Energy spectrum in T799 run
E(n)
)(log10 n
3/5k
3k
n = spherical harmonic order
missing energy
3/5k
3k
Lorentz Centre 2 October, 2006
ECMWF Model Output
• Shutts (2005) proposed a stochastic energy backscattering approach to compensate for the overdamping.
Lorentz Centre 2 October, 2006
ECMWF Model Output
• Shutts (2005) proposed a stochastic energy backscattering approach to compensate for the overdamping.
• His modifications allow for a substantially higher amount of energy at smaller scales.
Lorentz Centre 2 October, 2006
ECMWF Model Output
• Shutts (2005) proposed a stochastic energy backscattering approach to compensate for the overdamping.
• His modifications allow for a substantially higher amount of energy at smaller scales.
• The backscatter approach does produce the spectral kink at the mesoscales.
Lorentz Centre 2 October, 2006
Energy spectrum in T799 run
E(n)
)(log10 n
3/5k
3k
n = spherical harmonic order
missing energy
3/5k
3k
Lorentz Centre 2 October, 2006
Energy spectrum in ECMWF model with backscatter
3k
T799
E(n)
)(log10 n
3k
3/5k
Lorentz Centre 2 October, 2006
Some Outstanding Issues
• Flux Variability– Direction of (-5/3) short-wave energy cascade– Dependence on convective activity
Lorentz Centre 2 October, 2006
Some Outstanding Issues
• Flux Variability– Direction of (-5/3) short-wave energy cascade– Dependence on convective activity
• Geographic Variability– Strong convective activity– Little data collated in tropical areas
Lorentz Centre 2 October, 2006
Some Outstanding Issues
• Is it not possible for both Energy and Enstrophy to flow in both directions?
Lorentz Centre 2 October, 2006
Some Outstanding Issues
• Is it not possible for both Energy and Enstrophy to flow in both directions?
• In an unbounded system, a “W-shaped spectrum” may arise.
Lorentz Centre 2 October, 2006
Some Outstanding Issues
• Is it not possible for both Energy and Enstrophy to flow in both directions?
• In an unbounded system, a “W-shaped spectrum” may arise.
• For an additive spectrum, dominance
will alternate between -5/3 and -3 terms.
Lorentz Centre 2 October, 2006
Some Outstanding Issues
• The validity of an additive spectrum
needs to be justified.
E(k) Ak 3 Bk 5
3